Question: Solve for $x$ and $y$ using elimination. $\begin{align*}9x+6y &= 1 \\ 8x+9y &= 7\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-3$ and the bottom equation by $2$ $\begin{align*}-27x-18y &= -3\\ 16x+18y &= 14\end{align*}$ Add the top and bottom equations. $-11x = 11$ Divide both sides by $-11$ and reduce as necessary. $x = -1$ Substitute $-1$ for $x$ in the top equation. $9( -1)+6y = 1$ $-9+6y = 1$ $6y = 10$ $y = \dfrac{5}{3}$ The solution is $\enspace x = -1, \enspace y = \dfrac{5}{3}$.